O‘zmu xabarlari Вестник нууз acta nuuz

O‘zMU xabarlari Вестник НУУз ACTA NUUz
 
FIZIKA 
3/2/1 2021 
- 357 -
− 
− 
→ 
≥ 
∼  
Now, in order to see the equation of motion of a charged particle in an electro- magnetic field, we use the following 
equation [17] 
𝑑𝑣
𝜇
𝑑𝜏
+ Г
𝜈𝜆
𝜇
𝑣
𝜈
𝑣
𝜆
=
𝑒
𝑚
𝐹
𝜇𝜈
𝑣
𝜈
, (10) 
where m and e mass and electric charge of the particle, υµ = dxµ/dτ is four-velocity of the particle, 
𝛤
𝜈
𝜇
are Christoffel 
symbols for given spacetime metric and τ is an arbitrary affine parameter. Since open field lines area (the polar cap) is small 
(with an angular size of
~5
0
), we can use the small angle (
𝜃 ≪ 1) approximation for the polar cap region. Eq.(10) can be 
rewritten as [17] 
𝑁
2
(𝑠,𝑐
13
,𝑐
14
)
𝑠
2
𝑑
𝑑𝑠
(𝑠
2 𝑑ф
𝑑𝑠
) −
𝑙(𝑙+1)
𝑠
2
ф =
𝐵
𝐵
0
(
𝑗
𝑣
− 𝐽̅) , 
𝑑
𝑑𝑠
(𝑁(𝑠, 𝑐
13
, 𝑐
14
)𝛾) =
1
𝑁
2
(𝑠,𝑐
13
,𝑐
14
)
𝑑ф
𝑑𝑠
, (11) 
here l is the multipole number and the Lorentz factor γ is defined as 
𝛾 ≡ 𝑣
𝜇
𝑢
𝜇
=
1
√1−𝑉
2
, (12) 
where uµ is the four-velocity of fiducial observer, 𝑉
𝑖
≡
𝑣
𝜇
ℎ
𝜇
𝑖
𝛾
is the 3-dimensional spatial velocity with the projection 
tensor hαβ gαβ + uαuβ, and for convenience we introduced the following normalized variables [17] 
𝑗 ≡ −
2𝜋𝑁(𝑠)𝜌(𝑠)
Ω𝐵(𝑠)
,
ф(𝑠) ≡
𝑒
𝑚
Ф(𝑠) , 
𝑗̅ ≡ −
2𝜋𝑁(𝑠)𝜌
𝐺𝐽
(𝑠)
Ω𝐵(𝑠)
,
𝑠 ≡ √
2Ω𝐵𝑒
𝑚𝑐
2
𝑟, 
where, 
𝐵 ≡ |𝐵| = 𝐵
𝑟̂
+ 𝑂(𝜃
2
), Φ is the scalar potential of the electromagnetic field and ρ is the charge density on the 
stellar surface. In order to clarify the relationship between 
𝑗 and ¯j at 𝜒 = 0, we expand 𝑗 in power series for small parameter 
𝑦 =
𝑟
𝑅
− 1 up to the first order in the following form [17] 
𝑗̅ ≃ 𝑗̅
𝑅
[1 + 𝑦
3𝜔
𝑅
Ω
(1 +
𝑐
14
−2𝑐
13
1−𝑐
13
)
−1
] , (13) 
where 
𝑗̅ = 1 −
𝜔
𝑅
Ω
(1 +
𝑐
14
−2𝑐
13
1−𝑐
13
)
−1
with 
𝜔
𝑅
= 𝜔(𝑟 = 𝑅) It is clear that the presence of non-zero MOG 
parameter provides additional radial dependence on 
𝑗̅ In Fig. 3 we have shown dependence of Lorentz γ-factor of the 
accelerating charged particles on dimensionless height (
𝑦 = 𝜂 − 1) for different values of
aether parameters c13 and c14, for both cases 
𝑗 = 0.99𝑗
𝐺𝐽
, and 
𝑗 = 1.01𝑗
𝐺𝐽
Lorentz γ-factor increase in the presence of 
the parameters c13 and c14. It is seen from the figure that the velocity of the charged particle is zero at closer points in the right 
panel of the figure, in GR case the particles up to a critical distance radiates electromagnetic radiation due to 
Figure 3: Numerical solution of Eq.(11) for the Lorentz factor γ of a neutron stars with mass M = 2km, radius R = 10km 
and rotation period P = 1ms for l = 2. As initial condition, we use
𝛾
𝑅
= 𝛾
(𝑟=𝑅
= 1.000001. The left panel corresponds 
𝑗 = 0.99𝑗
𝐺𝐽
, the right one for
𝑗 = 1.01𝑗
𝐺𝐽
, 
inverse Compton scattering. In this scenario, the particle accelerates and due to the radiation it loses its kinetic energy. 
Then again accelerates by electric field parallel to the magnetic field lines, in turn the accelerated particles radiates again and 
again. In the distances over than the critic distance, the curvature radiation dominates. The Lorentz factor reaches up to about 60, 
it means that (
50 ≲ 𝛾 ≲ 104) the regime corresponds to resonance in Compton scattering in which the scattering cross-section 
increases significantly and thus the energy loss of the accelerated charged particle is high. Then the accelerated 
electrons/positrons radiate γ-rays through inverse Comp- ton scattering with the energy enough to create e± pairs, i.e.
𝐸
𝛾
≥
2𝑚
𝑒
𝑐
2
, in turn, the γ-rays start pair production 
𝛾 → 𝑒
+
+ 𝑒
−
in the magnetic field of the star. In the presence of aether 
parameter, the accelerated electrons from the surface of the star radiate at the distances closer to the surface than in general 
relativity and pair production process starts much earlier, because the charged particles are accelerating under two different 
forces: those of the accelerating electric field and the gravita- tional field of the central object. Here, we test the effects of the 
aether parameter on γ factor for the accelerating charged particle in a typical neutron star’s polar cap region. In
𝑗 > 𝑗̅ case (the 
bottom panel in figure 3) the value of the Lorentz factor at large distances reaches to more than 106 and the curvature radiation 
dominates being a source of ultra-high/high energy γ photons. 
Conclusions 
The present work, dedicated to acceleration processes around charged black holes with linear electric charge and rotating 
highly magnetized neutron stars, to understand the origin of high energetic protons, electrons and γ-rays, in cosmic ray spectrum. 
Energy release from charged black holes by electric Penrose process have been inves- tigated and shown that.... Finally, we also 
have studied energetic processes on the polar cap region of rotating magnetized neutron stars by inverse Compton scattering and 
curvature radiation processes. It is obtained that the presence of both aether parameters cause to increase the γ-factor of 
accelerating charged particles in both inverse Compton scattering and curvature radiation mechanisms. However, in curva- ture 
radiation case the value of relativistic factor about 
~10
6
times higher than it is in the case of inverse Compton scattering. 

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